Consider any locally checkable labeling problem \(\Pi \) in rooted regular trees: there is a finite set of labels \(\Sigma \), and for each label \(x \in \Sigma \) we specify what are permitted label combinations of the children for an internal node of label x (the leaf nodes are unconstrained). This formalism is expressive enough to capture many classic problems studied in distributed computing, including vertex coloring, edge coloring, and maximal independent set. We show that the distributed computational complexity of any such problem \(\Pi \) falls in one of the following classes: it is O(1), \(\Theta (\log ^* n)\), \(\Theta (\log n)\), or \(n^{\Theta (1)}\) rounds in trees with n nodes (and all of these classes are nonempty). We show that the complexity of any given problem is the same in all four standard models of distributed graph algorithms: deterministic \(\mathsf {LOCAL}\), randomized \(\mathsf {LOCAL}\), deterministic \(\mathsf {CONGEST}\), and randomized \(\mathsf {CONGEST}\) model. In particular, we show that randomness does not help in this setting, and the complexity class \(\Theta (\log \log n)\) does not exist (while it does exist in the broader setting of general trees). We also show how to systematically determine the complexity class of any such problem \(\Pi \), i.e., whether \(\Pi \) takes O(1), \(\Theta (\log ^* n)\), \(\Theta (\log n)\), or \(n^{\Theta (1)}\) rounds. While the algorithm may take exponential time in the size of the description of \(\Pi \), it is nevertheless practical: we provide a freely available implementation of the classifier algorithm, and it is fast enough to classify many problems of interest.

考虑有根正则树中的任何局部可检查标签问题\(\Pi \)：有一组有限标签\(\Sigma \)，对于每个标签\(x \in \Sigma \)我们指定允许的标签标签x的内部节点的子节点组合（叶节点不受约束）。这种形式主义的表现力足以捕捉分布式计算中研究的许多经典问题，包括顶点着色、边缘着色和最大独立集。我们证明任何此类问题的分布式计算复杂度\(\Pi \)属于以下类别之一：它是O (1), \(\Theta (\log ^* n)\) ,\(\Theta (\log n)\)或\(n^{\Theta (1)}\)在具有n 个节点的树中循环（并且所有这些类都是非空的）。我们表明，任何给定问题的复杂性在分布式图算法的所有四个标准模型中都是相同的：确定性\(\mathsf {LOCAL}\)、随机化\(\mathsf {LOCAL}\)、确定性\(\mathsf { CONGEST}\)和随机\(\mathsf {CONGEST}\)模型。特别是，我们表明随机性在此设置中没有帮助，复杂性类\(\Theta (\log \log n)\)不存在（虽然它确实存在于更广泛的一般树环境中）。我们还展示了如何系统地确定任何此类问题的复杂度等级 \(\Pi \)，即\(\Pi \)是否取O (1)，\(\Theta (\log ^* n)\)，\(\Theta (\log n)\)或\(n^{\Theta (1)}\)轮次。虽然该算法可能在\(\Pi \)的描述大小上花费指数级时间，但它仍然是实用的：我们提供了分类器算法的免费实现，它足够快来分类许多感兴趣的问题。